Gaussian radial-basis functions:: Cardinal interpolation of lp and power-growth data

Suppose lambda is a positive number. Basic theory of cardinal interpolation ensures the existence of the Gaussian cardinal function L-lambda(x) = Sigma(k is an element of Z) c(k) exp(-lambda(x-k)(2)), x is an element of R, satisfying the interpolatory conditions L-lambda(j)=delta(0j), j is an element of Z. The paper considers the Gaussian cardinal interpolation operator (L(lambda)y)(x):= Sigma(k is an element of Z) yk L-lambda(x - k), y = (y(k))(k is an element of Z), x is an element of R, as a linear mapping from l(p)(Z) into L-p(R), 1 less than or equal to p < infinity, and in particular, its behaviour as lambda --> 0(+). It is shown that parallel to L(lambda)parallel to(p) is uniformly bounded (in lambda) for 1<p < infinity, and that parallel to L(lambda)parallel to 1 log(1/lambda) as lambda --> 0(+). The limiting behaviour is seen to be that of the classical Whittaker operator W:y --> Sigma(k is an element of Z) y(k) sin pi(x-k)/pi(x-k), in that lim(lambda-->0+p) parallel to L(lambda)y - Wy parallel to(p) = 0, for every y is an element of l(p)(Z) and 1< p < infinity. It is further shown that the Gaussian cardinal interpolants to a function f which is the Fourier transform of a tempered distribution supported in (-pi,pi) converge locally uniformly to f as lambda --> 0(+). Multidimensional extensions of these results are also discussed.